Diffeomorphisms on Fuzzy Sphere

Abstract

Diffeomorphisms can be seen as automorphisms of the algebra of functions. In the matrix regularization, functions on a smooth compact manifold are mapped to finite size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through the matrix regularization. For the case of the fuzzy $S^2$, we construct the matrix regularization in terms of the Berezin-Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$N$ limit) under the general diffeomorphisms.